MATH 2000 - Introduction to Linear Algebra

Credit Hours: 3.00

Objectives: Students will:

1. Solve linear systems by elimination.
2. Describe solutions of systems geometrically and algebraically.
3. Define and reduce matrices to reduced row echelon form.
4. Use the Gaussian elimination to solve linear systems by forming an augmented matrix.
5. Use the Gauss-Jordan elimination to solve homogeneous and non homogeneous systems.

Outcome 2: Upon completion of this course the student will solve problems, use algorithms and demonstrate comprehension of concepts from: Matrices and their operations.

Objectives: Students will:

1. Perform the matrix operation.
2. Work with the properties of matrices.
3. Calculate the inverse of a matrix by using a formula or by reducing a n x 2n matrix.
4. Determine the operations on elementary matrices.
5. Solve linear system Ax = b by using A^-1 .

Outcome 3: Upon completion of this course the student will solve problems, use algorithms and demonstrate comprehension of concepts from: Determinants and their properties.

Objectives: Students will:

1. Evaluate the determinant of a matrix.
2. Use determinant to determine whether or not the inverse of a matrix exists.
3. Use elementary operation to evaluate determinants.
4. Work with the properties of determinants.
5. Use Cramer’s rule and determine to solve systems of linear equations of the form Ax = b, |A| not equal to 0.

Outcome 4: Upon completion of this course the student will solve problems, use algorithms and demonstrate comprehension of concepts from: Vector space.

Objectives: Students will:

1. Test sets for a vector space.
2. Establish a list of vector properties
3. Test a subset for a subspace of a vector space.
4. Test a solution vector of a homogeneous system to be a subspace of Rⁿ
5. Define a linear combination of vectors v₁, v₂,…v_n.
6. Test for spaces that are spanned by vectors.
7. Test for linearly independent and linearly dependent sets.
8. Determine whether a set is a basis for Rⁿ
9. Test for bases of a vector space.
10. Determine the dimension of a vector space and a solution space.
11. Find bases for a row space, column space and null space.

Outcome 5: Upon completion of this course the student will solve problems, use algorithms and demonstrate comprehension of concepts from: Inner product spaces.

Objectives: Students will:

1. Define Euclidean inner product R², on R³ and Rⁿ
2. Define the length and distance in the inner product space.
3. Define the norm in Rⁿ
4. Calculate an angle in the inner product space.
5. Establish the orthogonality in the inner product space.
6. Use the Gram-Schmidt process to find orthonormal bases for the inner product space.
7. Test for linear independent of orthogonal set in the inner space.
8. Find the coordinates of a vector v relative to orthonormal bases, and express v in terms of orthonormal bases.
9. Define orthogonal matrices.

Outcome 6: Upon completion of this course the student will solve problems, use algorithms and demonstrate comprehension of concepts from: Eigenvalues and eigen vectors of a matrix.

Objectives: Students will:

1. Find the eigenvalues and eigenvectors of a matrix.
2. Find the eigenvalues and eigenvectors of a powers of a matrix.
3. Test for diagonalizable matrices.
4. Define similar matrices.
5. Discuss the properties of a similar matrix.
6. Compute powers of a matrix.
7. Discuss orthogonal diagonalization of a symmetric matrix.
8. Use the eigenvalues and eigenvectors to express a general conic in standard form; find the angle of rotation and sketch.

Outcome 7: Upon completion of this course the student will solve problems, use algorithms and demonstrate comprehension of concepts from: Linear Transformations.

Objectives: Students will:

1. Define Linear Transformation T: V –> W
2. Discuss zero and identity transformations
3. Determine whether or not a transformation is linear
4. Find the standard matrix of a linear transformation
5. Find the Kernel and range of a linear transformation
6. Determine the rank and nullity of a linear transformation
7. Find the change of bases
8. Find nonstandard matrices of a linear transformation
9. Find matrices of sum, product, composition, and inverse of a linear transformation

1. Systems of Linear Equations
   1. Introduction to Systems of Linear Equations
      1. Gaussian Elimination and Gauss-Jordan Elimination
      2. Solving Systems of Homogeneous Linear Equations
      3. Solve Applications
   2. Matrices
      1. Operations with Matrices
      2. Properties of Matrix Operations
      3. The Inverse of a Matrix
      4. Elementary Matrices
      5. Solve Systems by Using Inverse Matrices
   3. Determinants
      1. The Determinant of a Matrix
      2. Evaluation of a Determinant Using Elementary Operations
      3. Properties of Determinants
      4. Solve Systems by Using Cramer’s Rule
      5. Use Determinants to find the Inverse of a Matrix
2. Vector space
   1. Vectors in Plane End Space
      1. Subspaces of Vector Spaces
      2. Spanning Sets and Linear Independence
      3. Basis and Dimension
      4. Rank of a Matrix and Systems of Linear Equations
      5. Coordinates and Change of Basis
      6. Applications of Vector Spaces
   2. Inner Product Spaces
      1. Length and Dot Product in Rn
      2. Inner Product Spaces
      3. Orthonormal Bases: Gram-Schmidt Process
      4. Applications of Inner Product Spaces
3. Eigenvalues and Eigenvectors
   1. Eigenvalues and Eigenvectors
   2. Diagonalization
   3. Symmetric Matrices and Orthogonal Diagonalization
   4. Use Eigenvalues and Eigenvectors to Rotate Axis and Sketch General Conic Sections
4. Linear Transformations
   1. Introduction to Linear Transformations
   2. The Kernel and Range of a Linear Transformation
   3. Standard and Non-Standard matrices for Linear Transformations
   4. Transition Matrices
   5. Applications of Linear Transformations